In the 1740s Squire published five essays on political subjects in which he voiced his support for the Whig party. His Letter to a Young Gentleman of Distinction (1740) argued for the benefits of a standing army against a militia to protect Britain in its wars against France and Spain. Squire advocated Britain's continental commitment of troops in The Important Question Discussed (1746). He came to the aid of Henry Pelham's administration by trumpeting its Whig principles in A Letter to a Tory Friend (1746). Squire also disputed the arguments of the Jacobite historian Thomas Carte by publishing two pamphlets in 1748: Remarks upon Mr. Carte's Specimen and A Letter to John Trot-Plaid. In the Remarks, Squire used natural law theory to contend against Carte's support of the House of Stuart, and in A Letter he satirised Carte by mocking his interpretation of the past in terms of the present.[2]

Squire also published two works on English history, An Enquiry into the Foundation of the English Constitution (1745) and Historical Essay upon the Balance of Civil Power in England (1748). In An Enquiry, Squire wrote on the German and Anglo-Saxon love of liberty and constitutionalism. In his Historical Essay, Squire wrote that liberty depended upon an equipoise among competing institutions and groups in society, suggesting that whenever such an equipoise collapses an arbitrary government takes its place. The Glorious Revolution of 1688 had ended the struggle to secure a balance and thus ensure liberty.[4]

In May 1746 Squire was elected a Fellow of the Royal Society as: "A Gentleman well known to the Learned World by Several valuable Treatises, perticularly [sic] 'Two Essays on the Antient Greek Chronology' and 'On the Origin of the Greek Language'; A new Edition of Plutarch's Discourse on 'Isis & Osiris', with an English Translation & Commentary; and an 'Historical Essay on the Anglo-Saxon Government both in Germany & England'".[5]